Reviewer:a e b
December 21, 2011 Subject:
Compact and useful
Ince, Ordinary Differential Equations, was published in 1926. It manages to pack a lot of good material into 528 pages. (With appendices it is 547 pages, but they are no longer relevant.) I have used Ince for several decades as a handy reference for Differential Equations. While it is primarily Ordinary Differential Equations (ODEs), it also has some material on Partial Differential Equations and Total Differential Equations. I like having an electronic copy of Ince as well as a hard copy, although some of the equations are slightly difficult to read on-line. If one wishes a single-volume work on ODEs, this is a good candidate.
Having said these positive things, I also must state that although Ince is an introductory book, it is not a wise choice for someone who has never encountered ODEs before. The compact presentation is great for anyone with familiarity, but the new-comer is well advised to get a slower-paced book. Also, for advanced results, Ince is too compact to have a lot of material. It is also too old to have modern work, such as on qualitative theory of ODEs. The book is reasonably free of typos, and those that do occur are trivial to spot and fix. For example, someone typed the original manuscript too fast 2/3 of the way down on page 49, typing "all planes which are parallel", where the logic requires "all planes that are perpendicular", and the following equation is for perpendicular.
I have several recent books on ODEs, but unless I am looking for something specific I know is in a particular book, I rarely consult them. Personally, I like A. R. Forsyth, The Theory of Differential Equations, best. It is 6 volumes, often grouped in pairs as 3 volumes, and several digital copies are available on Internet Archive. It has a more in-depth treatment than Ince, and (in my opinion) better proofs. For example, Forsyth gives a full volume to Pfaff's problem, whereas Ince devotes only a few pages. It is, of course a bit over a century old. But mathematics never becomes false over time. There are also modern treatments of ODEs that are excellent.
But again, if you want just one volume on ODEs, this is a good choice.